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Theorem prdssca 16469
Description: Scalar ring of a structure product. (Contributed by Stefan O'Rear, 5-Jan-2015.) (Revised by Mario Carneiro, 15-Aug-2015.) (Revised by Thierry Arnoux, 16-Jun-2019.)
Hypotheses
Ref Expression
prdsbas.p 𝑃 = (𝑆Xs𝑅)
prdsbas.s (𝜑𝑆𝑉)
prdsbas.r (𝜑𝑅𝑊)
Assertion
Ref Expression
prdssca (𝜑𝑆 = (Scalar‘𝑃))

Proof of Theorem prdssca
Dummy variables 𝑎 𝑐 𝑑 𝑒 𝑓 𝑔 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prdsbas.p . . . 4 𝑃 = (𝑆Xs𝑅)
2 eqid 2825 . . . 4 (Base‘𝑆) = (Base‘𝑆)
3 eqidd 2826 . . . 4 (𝜑 → dom 𝑅 = dom 𝑅)
4 eqidd 2826 . . . 4 (𝜑X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) = X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)))
5 eqidd 2826 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥)))))
6 eqidd 2826 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥)))))
7 eqidd 2826 . . . 4 (𝜑 → (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))) = (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥)))))
8 eqidd 2826 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥))))))
9 eqidd 2826 . . . 4 (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (∏t‘(TopOpen ∘ 𝑅)))
10 eqidd 2826 . . . 4 (𝜑 → {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))} = {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))})
11 eqidd 2826 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < )))
12 eqidd 2826 . . . 4 (𝜑 → (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))) = (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥))))
13 eqidd 2826 . . . 4 (𝜑 → (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))) = (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥))))))
14 prdsbas.s . . . 4 (𝜑𝑆𝑉)
15 prdsbas.r . . . 4 (𝜑𝑅𝑊)
161, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15prdsval 16468 . . 3 (𝜑𝑃 = (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩})))
17 eqid 2825 . . 3 (Scalar‘𝑃) = (Scalar‘𝑃)
18 scaid 16373 . . 3 Scalar = Slot (Scalar‘ndx)
19 elex 3429 . . . 4 (𝑆𝑉𝑆 ∈ V)
2014, 19syl 17 . . 3 (𝜑𝑆 ∈ V)
21 snsstp1 4565 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}
22 ssun2 4004 . . . . 5 {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
2321, 22sstri 3836 . . . 4 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩})
24 ssun1 4003 . . . 4 ({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2523, 24sstri 3836 . . 3 {⟨(Scalar‘ndx), 𝑆⟩} ⊆ (({⟨(Base‘ndx), X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))⟩, ⟨(+g‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(+g‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(.r‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(.r‘(𝑅𝑥))(𝑔𝑥))))⟩} ∪ {⟨(Scalar‘ndx), 𝑆⟩, ⟨( ·𝑠 ‘ndx), (𝑓 ∈ (Base‘𝑆), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑥 ∈ dom 𝑅 ↦ (𝑓( ·𝑠 ‘(𝑅𝑥))(𝑔𝑥))))⟩, ⟨(·𝑖‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑆 Σg (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(·𝑖‘(𝑅𝑥))(𝑔𝑥)))))⟩}) ∪ ({⟨(TopSet‘ndx), (∏t‘(TopOpen ∘ 𝑅))⟩, ⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ∧ ∀𝑥 ∈ dom 𝑅(𝑓𝑥)(le‘(𝑅𝑥))(𝑔𝑥))}⟩, ⟨(dist‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ sup((ran (𝑥 ∈ dom 𝑅 ↦ ((𝑓𝑥)(dist‘(𝑅𝑥))(𝑔𝑥))) ∪ {0}), ℝ*, < ))⟩} ∪ {⟨(Hom ‘ndx), (𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))⟩, ⟨(comp‘ndx), (𝑎 ∈ (X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) × X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥))), 𝑐X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ (𝑑 ∈ (𝑐(𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))(2nd𝑎)), 𝑒 ∈ ((𝑓X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)), 𝑔X𝑥 ∈ dom 𝑅(Base‘(𝑅𝑥)) ↦ X𝑥 ∈ dom 𝑅((𝑓𝑥)(Hom ‘(𝑅𝑥))(𝑔𝑥)))‘𝑎) ↦ (𝑥 ∈ dom 𝑅 ↦ ((𝑑𝑥)(⟨((1st𝑎)‘𝑥), ((2nd𝑎)‘𝑥)⟩(comp‘(𝑅𝑥))(𝑐𝑥))(𝑒𝑥)))))⟩}))
2616, 17, 18, 20, 25prdsvallem 16467 . 2 (𝜑 → (Scalar‘𝑃) = 𝑆)
2726eqcomd 2831 1 (𝜑𝑆 = (Scalar‘𝑃))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1656  wcel 2164  wral 3117  Vcvv 3414  cun 3796  wss 3798  {csn 4397  {cpr 4399  {ctp 4401  cop 4403   class class class wbr 4873  {copab 4935  cmpt 4952   × cxp 5340  dom cdm 5342  ran crn 5343  ccom 5346  cfv 6123  (class class class)co 6905  cmpt2 6907  1st c1st 7426  2nd c2nd 7427  Xcixp 8175  supcsup 8615  0cc0 10252  *cxr 10390   < clt 10391  ndxcnx 16219  Basecbs 16222  +gcplusg 16305  .rcmulr 16306  Scalarcsca 16308   ·𝑠 cvsca 16309  ·𝑖cip 16310  TopSetcts 16311  lecple 16312  distcds 16314  Hom chom 16316  compcco 16317  TopOpenctopn 16435  tcpt 16452   Σg cgsu 16454  Xscprds 16459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-oadd 7830  df-er 8009  df-map 8124  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-sup 8617  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-hom 16329  df-cco 16330  df-prds 16461
This theorem is referenced by:  pwssca  16509  xpssca  16591  xpsvsca  16592  prdslmodd  19328  dsmmlss  20451  rrxsca  23564
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